- ... filters,
^{1} - Following previous usage
conventions, we use the term ``ladder filter'' when the graph of the
filter is planar, and ``lattice filter'' when the graph is
non-planar.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
tube,
^{2} - ``Stiffness''
is defined here for air as the reciprocal of the adiabatic
compressibility of the gas [61, p. 230].
This definition helps to unify the
scattering formalism for acoustic tubes with that of mechanical systems
such as vibrating strings.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{3} - There are no tube shapes supporting exact
traveling waves other than cylindrical and conical (or conical wedge, which
is a hybrid) [69]. However, the ``Salmon horn family'' (see,
e.g., [60,96]) characterizes a larger class of
*approximate*``one-parameter traveling waves.'' In the cone, the wave equation is solved for pressure using a change of variables , where is the distance from the apex of the cone, causing the wave equation for the cone to reduce to that of the cylindrical case [2].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... power
^{4} - Note that
corresponds to the
average physical power at frequency , where
,
and the wave variable magnitudes on the unit circle may be interpreted as
RMS levels [4, p. 48]. For , we may interpret the
power
as the steady state power obtained when
exponential damping is introduced into the waveguide giving decay
time-constant , where
[4, p.
48].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{5} - A complex-valued function
of a complex variable is said to be
*positive real*if- 1)
- 2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
^{6} - Note that by the maximum modulus theorem, if a
matrix of meromorphic functions
is positive definite on the unit circle and analytic outside the unit circle,
it is necessarily positive definite everywhere outside the unit circle.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... velocity
^{7} - The symbols for the variables
velocity and force have been chosen to maintain consistency with the
analogous acoustical quantities.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... sample,
^{8} - Strictly speaking, a
squared wave variable in a normalized waveguide is proportional to signal
*power*[97], and we should multiply by the sampling interval to obtain units of energy. However, such a constant scale factor is inconsequential.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...WAVCAS.
^{9} - Altering a ladder filter to convert it
to a DWN can also be accomplished using the cut-set method of Kung
[51,12].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .